3.2.14 \(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^4} \, dx\) [114]

3.2.14.1 Optimal result

Integrand size = 28, antiderivative size = 1007 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=-\frac {160 b^3 f m n^3}{27 e x}-\frac {4 b^3 f^{3/2} m n^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 e^{3/2}}-\frac {52 b^2 f m n^2 \left (a+b \log \left (c x^n\right )\right )}{9 e x}-\frac {4 b^2 f^{3/2} m n^2 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 e^{3/2}}-\frac {8 b f m n \left (a+b \log \left (c x^n\right )\right )^2}{3 e x}-\frac {2 f m \left (a+b \log \left (c x^n\right )\right )^3}{3 e x}+\frac {b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {f^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b^2 f^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{(-e)^{3/2}}+\frac {2 b^2 f^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {b f^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{(-e)^{3/2}}+\frac {2 i b^3 f^{3/2} m n^3 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}-\frac {2 i b^3 f^{3/2} m n^3 \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 e^{3/2}}+\frac {2 b^3 f^{3/2} m n^3 \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}+\frac {2 b^2 f^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{(-e)^{3/2}}-\frac {2 b^3 f^{3/2} m n^3 \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 (-e)^{3/2}}-\frac {2 b^2 f^{3/2} m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{(-e)^{3/2}}-\frac {2 b^3 f^{3/2} m n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{(-e)^{3/2}}+\frac {2 b^3 f^{3/2} m n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{(-e)^{3/2}} \]

output

-160/27*b^3*f*m*n^3/e/x-8/3*b*f*m*n*(a+b*ln(c*x^n))^2/e/x-52/9*b^2*f*m*n^2 
*(a+b*ln(c*x^n))/e/x-2/9*b^2*n^2*(a+b*ln(c*x^n))*ln(d*(f*x^2+e)^m)/x^3-1/3 
*b*n*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m)/x^3+1/3*f^(3/2)*m*(a+b*ln(c*x^n)) 
^3*ln(1-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-1/3*f^(3/2)*m*(a+b*ln(c*x^n))^3*l 
n(1+x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-b*f^(3/2)*m*n*(a+b*ln(c*x^n))^2*polyl 
og(2,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+b*f^(3/2)*m*n*(a+b*ln(c*x^n))^2*pol 
ylog(2,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2/9*I*b^3*f^(3/2)*m*n^3*polylog(2, 
-I*x*f^(1/2)/e^(1/2))/e^(3/2)-2/27*b^3*n^3*ln(d*(f*x^2+e)^m)/x^3-1/3*(a+b* 
ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^3-4/27*b^3*f^(3/2)*m*n^3*arctan(x*f^(1/2) 
/e^(1/2))/e^(3/2)+2/3*b^3*f^(3/2)*m*n^3*polylog(3,-x*f^(1/2)/(-e)^(1/2))/( 
-e)^(3/2)-2/3*b^3*f^(3/2)*m*n^3*polylog(3,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2) 
-2*b^3*f^(3/2)*m*n^3*polylog(4,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2*b^3*f^( 
3/2)*m*n^3*polylog(4,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-4/9*b^2*f^(3/2)*m*n^ 
2*arctan(x*f^(1/2)/e^(1/2))*(a+b*ln(c*x^n))/e^(3/2)+1/3*b*f^(3/2)*m*n*(a+b 
*ln(c*x^n))^2*ln(1-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-1/3*b*f^(3/2)*m*n*(a+b 
*ln(c*x^n))^2*ln(1+x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2/3*b^2*f^(3/2)*m*n^2* 
(a+b*ln(c*x^n))*polylog(2,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2/3*b^2*f^(3/2 
)*m*n^2*(a+b*ln(c*x^n))*polylog(2,x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)+2*b^2*f 
^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3,-x*f^(1/2)/(-e)^(1/2))/(-e)^(3/2)-2 
*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x^n))*polylog(3,x*f^(1/2)/(-e)^(1/2))/(-e)...
 

3.2.14.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2488\) vs. \(2(1007)=2014\).

Time = 0.56 (sec) , antiderivative size = 2488, normalized size of antiderivative = 2.47 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Result too large to show} \]

input

Integrate[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^4,x]
 

output

(-18*a^3*Sqrt[e]*f*m*x^2 - 72*a^2*b*Sqrt[e]*f*m*n*x^2 - 156*a*b^2*Sqrt[e]* 
f*m*n^2*x^2 - 160*b^3*Sqrt[e]*f*m*n^3*x^2 - 18*a^3*f^(3/2)*m*x^3*ArcTan[(S 
qrt[f]*x)/Sqrt[e]] - 18*a^2*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
- 12*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*b^3*f^(3/2)*m 
*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 54*a^2*b*f^(3/2)*m*n*x^3*ArcTan[(Sq 
rt[f]*x)/Sqrt[e]]*Log[x] + 36*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/S 
qrt[e]]*Log[x] + 12*b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[ 
x] - 54*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 18* 
b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 18*b^3*f^(3/2 
)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 - 54*a^2*b*Sqrt[e]*f*m*x^ 
2*Log[c*x^n] - 144*a*b^2*Sqrt[e]*f*m*n*x^2*Log[c*x^n] - 156*b^3*Sqrt[e]*f* 
m*n^2*x^2*Log[c*x^n] - 54*a^2*b*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]* 
Log[c*x^n] - 36*a*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^ 
n] - 12*b^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 108 
*a*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 36* 
b^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 54*b 
^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] - 54* 
a*b^2*Sqrt[e]*f*m*x^2*Log[c*x^n]^2 - 72*b^3*Sqrt[e]*f*m*n*x^2*Log[c*x^n]^2 
 - 54*a*b^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - 18*b^ 
3*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + 54*b^3*f^(...
 

3.2.14.3 Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 989, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx\)

\(\Big \downarrow \) 2825

\(\displaystyle -2 f m \int \left (-\frac {2 b^3 n^3}{27 x^2 \left (f x^2+e\right )}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{9 x^2 \left (f x^2+e\right )}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 n}{3 x^2 \left (f x^2+e\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 x^2 \left (f x^2+e\right )}\right )dx-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b^3 \log \left (d \left (f x^2+e\right )^m\right ) n^3}{27 x^3}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) n^2}{9 x^3}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) n}{3 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right )}{3 x^3}-2 f m \left (\frac {2 b^3 \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) n^3}{27 e^{3/2}}-\frac {i b^3 \sqrt {f} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{9 e^{3/2}}+\frac {i b^3 \sqrt {f} \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{9 e^{3/2}}-\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{3 (-e)^{3/2}}+\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{3 (-e)^{3/2}}+\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{(-e)^{3/2}}-\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{(-e)^{3/2}}+\frac {80 b^3 n^3}{27 e x}+\frac {2 b^2 \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right ) n^2}{9 e^{3/2}}+\frac {26 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{9 e x}+\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{3 (-e)^{3/2}}-\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{3 (-e)^{3/2}}-\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{(-e)^{3/2}}+\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{(-e)^{3/2}}+\frac {4 b \left (a+b \log \left (c x^n\right )\right )^2 n}{3 e x}-\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{6 (-e)^{3/2}}+\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) n}{6 (-e)^{3/2}}+\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 (-e)^{3/2}}-\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 (-e)^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 e x}-\frac {\sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{6 (-e)^{3/2}}+\frac {\sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{6 (-e)^{3/2}}\right )\)

input

Int[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^4,x]
 

output

(-2*b^3*n^3*Log[d*(e + f*x^2)^m])/(27*x^3) - (2*b^2*n^2*(a + b*Log[c*x^n]) 
*Log[d*(e + f*x^2)^m])/(9*x^3) - (b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^ 
2)^m])/(3*x^3) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(3*x^3) - 2*f 
*m*((80*b^3*n^3)/(27*e*x) + (2*b^3*Sqrt[f]*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
)/(27*e^(3/2)) + (26*b^2*n^2*(a + b*Log[c*x^n]))/(9*e*x) + (2*b^2*Sqrt[f]* 
n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)) + (4*b*n*( 
a + b*Log[c*x^n])^2)/(3*e*x) + (a + b*Log[c*x^n])^3/(3*e*x) - (b*Sqrt[f]*n 
*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(6*(-e)^(3/2)) - (Sqr 
t[f]*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(6*(-e)^(3/2)) + 
(b*Sqrt[f]*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(6*(-e)^( 
3/2)) + (Sqrt[f]*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(6*(- 
e)^(3/2)) + (b^2*Sqrt[f]*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/S 
qrt[-e])])/(3*(-e)^(3/2)) + (b*Sqrt[f]*n*(a + b*Log[c*x^n])^2*PolyLog[2, - 
((Sqrt[f]*x)/Sqrt[-e])])/(2*(-e)^(3/2)) - (b^2*Sqrt[f]*n^2*(a + b*Log[c*x^ 
n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (b*Sqrt[f]*n*(a + b 
*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(2*(-e)^(3/2)) - ((I/9)*b 
^3*Sqrt[f]*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) + ((I/9)*b^3* 
Sqrt[f]*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (b^3*Sqrt[f]*n^3* 
PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (b^2*Sqrt[f]*n^2*(a 
+ b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) + (b^3*...
 

3.2.14.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2825

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* 
(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r 
 Int[x^(m - 1)/(e + f*x^m)   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

3.2.14.4 Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{4}}d x\]

input

int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^4,x)
 

output

int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^4,x)
 

3.2.14.5 Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}} \,d x } \]

input

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="fricas")
 

output

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a 
^3)*log((f*x^2 + e)^m*d)/x^4, x)
 

3.2.14.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Timed out} \]

input

integrate((a+b*ln(c*x**n))**3*ln(d*(f*x**2+e)**m)/x**4,x)
 

output

Timed out
 

3.2.14.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]

input

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.2.14.8 Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}} \,d x } \]

input

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="giac")
 

output

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + e)^m*d)/x^4, x)
 

3.2.14.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^4} \,d x \]

input

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^3)/x^4,x)
 

output

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^3)/x^4, x)